概率与统计(英文)chapter 2 probability

2probability2.1SampleSpaceandEvents2.2Axioms,Interpretations,andPropertiesofProbability2.3CountingTechniques2.4ConditionalProbability2.5IndependenceIntroduction

Thetermprobabilityreferstothestudyofrandomnessanduncertainty.Inanysituationinwhichoneofanumberofpossibleoutcomesmayoccur,thetheoryofprobabilityprovidesmethodsforquantifyingthechances,orlikelihoods,associatedwiththevariousoutcomes.Thelanguageofprobabilityisconstantlyusedinaninformalmannerinbothwrittenandspokencontexts.Inthischapter,weintroducesomeelementaryprobabilityconcepts,indicatehowprobabilitiescanbeinterpreted,andshowhowtherulesofprobabilitycanbeappliedtocomputetheprobabilitiesofmanyinterestingevents.Themethodologyofprobabilitywillthenpermitustoexpressinpreciselanguagesuchinformalstatementsasthosegivenabove.2.1SampleSpacesandEventsAnexperimentisanyactionorprocessthatgeneratesobservations.Forexamples,tossingacoinonceorseveraltimes,selectingacardorcardsfromadeck,weighingaloafofbread,ascertainingthecommutingtimefromhometoworkonparticularmorning,obtainingbloodtypesfromagroupofindividuals,etc.RandomexperimentSamplespaceandsamplepointDefinition:Thesamplespaceofanexperiment,denotedbyS,orΩ,isthesetofallpossibleoutcomesofthatexperiment,andsamplepointofthesamplespace,denotedbys,isaoutcomeoftheexperiment.Example2.1:Thesimplestexperimenttowhichprobabilityappliesisonewithtwopossibleoutcomes.Onesuchexperimentconsistsofexaminingasinglefusetoseewhetheritisdefective.ThesamplespaceforthisexperimentcanbeabbreviatedasS={N,D},whereNrepresentsnotdefective,Drepresentsdefective,andthebracesareusedtoenclosetheelementsofaset.Anothersuchexperimentwouldinvolvetossingathumbtackandnotingwhetheritlandedpointuporpointdown,withsamplespaceS

={U,D},andyetanotherwouldconsistofobservingthesexofthenextchildbornatthelocalhospital,withS

={M,F}.Example2.2:Ifweexaminethreefusesinsequenceandnotetheresultofeachexamination,thenanoutcomefortheentireexperimentisanysequenceofN’sandD’soflength3,soS={NNN,NND,NDN,NDD,DNN,DND,DDN,DDD}.Example2.2:drivingtowork,acommuterpassesthroughasequenceofthreeintersectionswithtrafficlights.Ateachlight,sheeitherstops,s,orcontinues,c,thesamplespaceisthesetofallpossibleoutcomes:

S={ccc,ccs,css,csc,sss,ssc,scc,scs}

Wherecscforexampledenotestheoutcomesthatthecommutercontinuesthroughthefirstlight,stopsatthesecondlight,andcontinuesthroughthethirdlight.Example2.3

Thenumberofjobsinaprintqueueofamainframecomputermaybemodeledasrandom.HerethesamplespacecanbetakenasS

={0,1,2,3,….}Thatis,allthenonnegativeintegers.Inpracticethereisprobablyanupperlimit,N,onhowlargetheprintqueuecanbe,soinsteadthesamplespacemightbedefinedas

S

={0,1,2,3,…,N}Example2.3Twogasstationsarelocatedatacertainintersection.Eachonehassixgaspumps.Considertheexperimentinwhichthenumberofpumpsinuseataparticulartimeofdayisdeterminedforeachofthestations.Anexperimentaloutcomespecifieshowmanypumpsareinuseatthefirststationandhowmanyareinuseatthesecondone.The49outcomesinSaredisplayedintheaccompanyingtable.

01234560(0,0)(0,1)(0,2)(0,3)(0,4)(0,5)(0,7)1(1,0)(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)2(2,0)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)3(3,0)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)4(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)5(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)6(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)Example2.4Ifanewtype-Dflashlightbatteryhasavoltagethatisoutsidecertainlimits,thatbatteryischaracterizedasafailure(F);ifthebatteryhasavoltagewithintheprescribedlimits,itisasuccess(S).Supposeanexperimentconsistsoftestingeachbatteryasitcomesoffanassemblylineuntilwefirstobserveasuccess.Althoughitmaynotbeverylikely,apossibleoutcomeofthisexperimentisthatthefirst10(or100or1000or…)aref’sandthenextoneisanS.thesamplespaceisΩ={S,FS,FFS,FFFS,…..}Example2.4Earthquakesexhibitedveryerraticbehavior,whichissometimesmodeledasrandom.Forexample,thelengthoftimebetweensuccessiveearthquakesinaparticularregionthataregreaterinmagnitudethanagiventhresholdmayberegardedasanexperiment.HereΩisthesetofallnonnegativenumbers:

Ω=RandomeventDefinition:Anevent,areusuallydenotedbyitalicuppercaseletters,isanycollection(subset)ofoutcomescontainedinthesamplespaceS

.Aneventissaidtobesimpleifitconsistsofexactlyoneoutcomeandcompoundifitconsistsofmorethanoneoutcome.Remark:Whenanexperimentisperformed,aparticulareventAissaidtooccuriftheresultingexperimentaloutcomeiscontainedinA.Ingeneral,exactlyonesimpleeventwilloccur,butmanycompoundeventswilloccursimultaneously.TheeventthatthecommuterstopsatthefirstlightisthesubsetofSdenotedbyA={sss,ssc,scc,scs}Example2.5:Consideranexperimentinwhicheachofthreeautomobilestakingaparticularfreewayexitturnsleft(L)orright(R)attheendoftheexitramp.TheeightpossibleoutcomesthatcomprisethesamplespaceareLLL,RLL,LRL,LLR,LRR,RLR,RRL,andRRR.Thus,thereareeightsimpleevents,amongwhichareE1={LLL}andE5={LRR}.SomecompoundeventsincludeA={RLL,LRL,LLR}=theeventthatexactlyoneofthethreecarsturnsrightB={LLL,RLL,LRL,LLR}=theeventthatatmostoneofthecarsturnsrightC={LLL,RRR}=theeventthatallthreecarsturninthesamedirectionSupposethatwhentheexperimentisperformed,theoutcomeisLLL.ThenthesimpleeventE1hasoccurredandsoalsohavetheeventsBandC(butnotA).Example2.6Whenthenumberofpumpsinuseateachoftwosix-pumpgasstationsisobserved,thereare49possibleoutcomesA={(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}=theeventthatthenumberofpumpsinuseisthesameforbothstationsB={(0,4),(1,3),(2,2),(3,1),(4,0)}=theeventthatthetotalnumberofpumpsinuseisfourC={(0,0),(0,1),(1,0),(1,1)}=theeventthatatmostonepumpisinuseateachstationExample2.7Thesamplespaceforthebatteryexaminationexperimentcontainsaninfinitenumberofoutcomes,sothereareaninfinitenumberofsimpleevents.Compoundeventsinclude

A={S,FS,FFS}=theeventthatatmostthreebatteriesareexaminedE={FS,FFFS,FFFFFS,….}=theeventthatanevennumberofbatteriesareexamined.ExerciseP57.2.Supposethatvehiclestakingaparticularfreewayexitcanturnright(R),turnleft(L),orgostraight(S).Considerobservingthedirectionforeachofthreesuccessivevehicles.ListalloutcomesintheeventAthatallthreevehiclesgointhesamedirection.ListalloutcomesintheeventBthatallthreevehiclestakedifferentdirections.ListalloutcomesintheeventCthatexactlytwoofthethreevehiclesturnright.ListalloutcomesintheeventDthatexactlytwovehiclesgointhesamedirection.Thealgebraofsettheorycarriesoverdirectlyintoprobabilitytheory.Theunionoftwoevents,AandB,istheeventCthateitherAoccursorBoccursorbothoccur:C=A∪B.Forexample,ifAistheeventthatcommuterstopsatthefirstlightandifBistheeventthatshestopsatthethirdlight,

Union(∪)ThenCistheeventthathestopsatthefirstlightorstopsatthethirdlightandconsistsoftheoutcomesthatareinAorinBorinboth:SomerelationsfromsettheoryAneventisnothingbutaset,sothatrelationshipsandresultsfromelementarysettheorycanbeusedtostudyevents.Thefollowingconceptsfromsettheorywillbeusedtoconstructneweventsfromgivenevents.Theintersectionoftwoevents,C=A∩B,istheeventthatbothAandBoccur.IfAandBareaslistedpreviously,thenCistheeventthatthecommuterstopsatthefirstlightandstopsatthethirdlightandthusconsistsofthoseoutcomesthatarecommontobothAandB:C=intersection(∩)Thecomplementofanevent,,istheeventthatAdoesnotoccurandthusconsistsofallthoseelementsinthesamplespacethatarenotinA,thecomplementoftheeventthatthecommuterstopsatthefirstlightistheeventthatshecontinuesatthefirstlight:=Complement(or)Definition:WhenAandBhavenooutcomesincommon,theyaresaidtobemutuallyexclusiveordisjointevents.Example2.10Asmallcityhasthreeautomobiledealerships:aGMdealersellingChevrolets,Pontiacs,andBuicks;aForddealersellingFordsandMercurys;andaChryslerdealersellingPlymouthsandChryslers.Ifanexperimentconsistsofobservingthebrandofthenextcarsold,thentheeventsA=(Chevrole,Pontiac,Buick}andB={Ford,Mercury}aremutuallyexclusivebecausethenextcarsoldcannotbebothaGMproductandaFordproduct.Youmayrecallfrompreviousexposuretosettheorythatrathermysterioussetcalledtheemptyset,usuallydenotedbyФ.Theemptysetwithnoelements;itistheeventwithnooutcomes.Forexample,ifAistheeventthatthecommuterstopsatthefirstlightandCistheeventthatshecontinuesthroughallthreelights,c=thenAandChavenooutcomesincommon,andwecanwrite:

Insuchcases,AandCaresaidtobedisjoint.ABΩABAABABA∩BBA－BAABVenndiagramsThefollowingaresomelowsofsettheory.Commutativelaws:Associativelaws:Distributivelaws:DeMorganlaws:ExerciseP572,4Exercise1SupposeA,B,Carerandomevents.PleasedescribethefollowingeventsbyA,B,C,andtheirsetrelations.(1)OnlyApresent(2)onlyAnotpresent(3)Thethreeeventsareoccuratthesametime(4)Noneofthethreeispresent(5)Onlyoneofthethreeeventsispresent(6)Twoofthethreeeventsarepresent(7)Atleastoneispresent(8)Atleasttwoarepresent(9)Atmosttwoarepresent(10)Atmostoneispresent(11)A,B,CarenotpresentatthesametimeExerciseP57.2.Supposethatvehiclestakingaparticularfreewayexitcanturnright(R),turnleft(L),orgostraight(S).Considerobservingthedirectionforeachofthreesuccessivevehicles.ListalloutcomesintheeventAthatallthreevehiclesgointhesamedirection.ListalloutcomesintheeventBthatallthreevehiclestakedifferentdirections.ListalloutcomesintheeventCthatexactlytwoofthethreevehiclesturnright.ListalloutcomesintheeventDthatexactlytwovehiclesgointhesamedirection.ListoutcomesinExerciseP57.4.Eachofasampleoffourhomemortgagesisclassifiedasfixedrate(F)orvariablerate(V).Whatarethe16outcomesinS?Whichoutcomesareintheeventthatexactlythreeoftheselectedmortgagesarefixedrate?Whichoutcomesareintheeventthatallfourmortgagesareofthesametype?Whichoutcomesareintheeventthatatmostoneofthefourisavariables-ratemortgage?Whatistheunionoftheeventsinpart(c)and(d),andwhatistheintersectionofthesetwoevents?Whataretheunionandintersectionofthetwoeventsinparts(b)andpart(c)?2.2Axioms,interpretationsandpropertiesofprobabilityGivenanexperimentandasamplespaceS,theobjectiveofprobabilityistoassigntoeacheventAanumberP(A),calledtheprobabilityoftheeventA,whichwillgiveaprecisemeasureofthechancethatAwilloccur.Toensurethattheprobabilityassignmentswillbeconsistentwithourintuitivenotionsofprobability,allassignmentsshouldsatisfythefollowingaxioms:3.IfA1andA2aredisjoint,thenThefirsttwoaxiomsareratherobvious.SinceSconsistsofallpossibleoutcomes.Thesecondaxiomsimplystatesthataprobabilityisnonnegative.

Moregenerally,ifaremutuallydisjoint,thenThethirdaxiomstatesthatifAandBaredisjointthatishavenooutcomesincommonthenandalsothatthispropertyextendstolimits.Forexampletheprobabilitythattheprintqueuecontainseitheroneorthreejobsisequaltotheprobabilitythatitcontainsoneplustheprobabilitythatitcontainsthree.Example2.12ConsidertheexperimentinExample2.4,inwhichbatteriescomingofanassemblylinearetestedonebyoneuntilonehavingavoltagewithinprescribedlimitsisfound.ThesimpleeventsareE1={S},E2={FS},E3=[FFS},E4={FFFS},….Supposetheprobabilityofanyparticularbatterybeingsatisfactoryis0.99.ThenThefollowingpropertiesofprobabilitymeasuresareconsequencesoftheaxioms.Property1thispropertyfollowssinceAandaredisjointwithandthus,bythefirstandthirdaxioms,inwordsthispropertysaysthattheprobabilitythataneventdoesnotoccurequalsoneminustheprobabilitythatitdoesoccur.Example2.13Considerasystemoffiveidenticalcomponentsconnectedinseries,asillustratedinthefollowingfigure12345DenoteacomponentthatfailsbyFandonethatdoesn’tfailbyS(forsuccess).Supposethattheprobabilityofonecomponentdoesn’tfailis0.9.LetAbetheeventthatthesystemfails.Property2

thispropertyfollowsfromproperty1sinceInwordsthissaysthattheprobabilitythatthereisnooutcomeatalliszero.Property3ifthenthispropertyfollowssinceBcanbeexpressedastheunionoftwodisjointsets:Thenfromthethirdaxiom,AndthusThispropertystatesthatifBoccurswheneverAoccurs,then.Forexampleifwheneveritrains(A)itiscloudy(B),thentheprobabilitythatitrainsislessthanorequaltotheprobabilitythatitiscloudy.Property4Additionlawtoseethis,wedecomposeintothreedisjointsubsets,asshowninfigure2.1:FIGURE2.1Venndiagramillustratingtheadditionlaw.

ECD

BAExample2.14Inacertainresidentialsuburb,60%ofallhousehokdssubscribetothemetropolitannewspaperpublishedinanearbycity,80%subscribetothelocalafternoonpaper,and50%ofallhouseholdssubscribetobothpapers.Ifahouseholdisselectedatrandom,whatistheprobabilitythatitsubscribesto(1)atleastoneofthetwonewspapersandwhat(2)exactlyoneofthetwonewspapers?Solution:WithA={subscribestothemetropolitanpaper}B={subscribestothelocalpaper}ThenP(A)=0.6,P(B)=0.8,andSo(1)P(subscribestoatleastoneofthetwonewspapers)=(2)P(exactlyone)=Theprobabilityofaunionofmorethantwoeventscanbecomputedanalogously.ForthreeeventsA,B,andC,theresultisExample:Twofightersshootaplane,theprobabilityofonefighterhittheplaneis0.4,theotheris0.6,andbothatthesametimehittheplaneprobabilityis0.3.Determinetheprobabilitythatatleastonefighterhittheplane?Example:Inarecentstudyonteenagedrugandalcoholuse,researchersfoundthatoneintenteenagersadmittedtousingmarijuanaatleastonceamonth,oneinfiveadmittedtodrinkingalcoholatleastonceaweek,andoneinfouradmittedtosmokingcigarettesonadailybasis.Theresearchersalsofoundthat10%ofrespondentsadmittedtobothsmokingcigarettesdailyanddrinkingalcoholatleastonceaweek;5%ofrespondentsadmittedtobothsmokingcigarettesdailyandusingmarijuanaatleastonceamonth;3%ofrespondentsadmittedtobothdrinkingalcoholatleastonceaweekandusingmarijuanaatleastonceamonth;and1%ofrespondentsadmittedtodoingallthree.Assumingthattheteenagerisaslikelyastheothersinthestudytosmoke,drink,ortakedrugs,whatistheprobabilitythattheteenagerengagesinatleastoneofthesethreepoorbehaviors?Whatistheprobabilitythattheteenagerdoesnotengageinanyofthesebehaviors?Solution:WesetM={Teenageradmitstousingmarijunanatleastonceamonth}A={Teenageradmitstoconsumingalcoholatleastonceaweek}C={Teenageradmitstosmokingcigareteesonadailybasis}So,P(M)=0.1,P(A)=0.2,P(C)=0.25ThenExample:Considerarecentstudyconductedbythepersonnelmanagerofamajorcomputersoftwarecompany.Itwasfoundthat30%oftheemployeeswholeftthefirmwithintwoyearsdidsoprimarilybecausetheyweredissatisfiedwiththeirsalary,20%leftbecausetheyweredissatisfiedwiththeirworkassignments,and12%oftheformeremployeesindicateddissatisfactionwithboththeirsalaryandtheirworkassignment.Whatistheprobabilitythatanemployeewholeaveswithintwoyearsdoessobecauseofdissatisfactionwithsalary,dissatisfactionwiththeworkassignment,orboth?

Solution:S={leavesbecauseofdissatisfactionwithsalary}W={leavesbecauseofdissatisfactionwithworkassignment}ExampleifP(A)=P(B)=P(C)=1/4,P(AB)=P(AC)=0,P(BC)=3/16,Determinetheprobabilityofevent={noneofA,B,Coccurs}ExampleSupposeP(A)=0.4,P(B)=0.3,then

Exercise3:WhenA,Boccuratthesametime,theneventCalsooccurs,then(A)P(C)=P(AB)(B)P(C)=P(A+B)Exercise4:A,B,andCare

mutuallyexclusiveevents,P(A)=0.2,P(B)=0.3,P(C)=0.4,thenP(A+B-C)=(A)0.1,(B)0.3(C)0.5(D)0.44EquallylikelyoutcomesInmanyexperimentsconsistingofNoutcomes,itisreasonabletoassignequalprobabilitiestoallNsimpleevents.Withp=P(Ei)foreveryi,Thatis,ifthereareNpossibleoutcomes,thentheprobabilityassignedtoeachis1/N.NowconsideraneventA,withN(A)denotingthenumberofoutcomescontainedinA.ThenExample2.6:Whentwodicearerolledseparately,thereareN=36outcomes.Ifboththedicearefair,all36outcomesareequallylikely,soP(Ei)=1/36.Otherquestion:A={thesumoftwodiceis7},determineP(A)?Therearetwoanswer,whichiscorrect?Answer1:thesamplespaceΩ={2,3,4,5,6,7,8,9,10,11,12},soP(A)=1/11Answer2:thesamplespaceisΩ={(1,1),(1,2),(1,3),(1,4)…..}SoP(A)=6/36=1/6ExerciseP65121314172.3CountingtechniquesWhenthevariousoutcomesofanexperimentareequallylikely,thenthetaskofcomputingprobabilitiesreducestocounting.Inparticular,ifNisthenumberofoutcomesinasamplespaceandN(A)isthenumberofoutcomescontainedinaneventA,then(2.1)IfalistoftheoutcomesisavailableoreasytoconstructandNissmall,thenthenumeratoranddenominatorofEquation(2.1)canbeobtainedwithoutthebenefitofanygeneralcountingprinciples.Example2.17:ablackurncontains5redand6greenballsandawhiteurncontains3redand4greenballs.Youareallowedtochooseanurnandthenchooseaballatrandomfromtheurn.Ifyouchoosearedball,yougetaprize.Whichurnshouldyouchoosetodrawfrom?Ifyoudrawfromtheblackurn,theprobabilityofchoosingaredballis0.455.Ifyouchoosetodrawfromthewhiteurn,theprobabilityofchoosingaredballis0.429,soyoushouldchoosetodrawfromtheblackurn.

Simpson’sparadox.Nowconsideranothergameinwhichasecondblackurnhas6redand3greenballsandasecondwhiteurnhas9redand5greenballs.Ifyoudrawfromtheblackurn,theprobabilityofredballis6/9.whereasifyouchoosetodrawfromthewhiteurn,theprobabilityis9/14.so,againyoushouldchoosetodrawfromtheblackurn.Inthefinalgame,thecontentsofthesecondblackurnareaddedtothefirstblackurnandthecontentsofthesecondwhiteurnareaddedtothefirstwhiteurn.againyoucanchoosewhichurnyoucandrawfrom.whichshouldyouchoose?Intuitionsayschoosetheblackurn,butlet’scalculatetheprobabilities.

Theblackurnnowcontains11redand9greenballs,sotheprobabilityofdrawingaredballfromitis11/20=0.55.Thewhiteurnnowcontains12redand9greenballs,sotheprobabilityofdrawingaredballfromitis12/21=0.571.

so,youshouldchoosethewhiteurn.ThiscounterintuitiveresultisanexampleofSimpson’sparadox.Theproductrulefororderedpairs

Ifoneexperimenthasmoutcomesandanotherexperimenthasnoutcomes,thentherearem×npossibleoutcomesforthetwoexperiments.Proof:DenotetheoutcomesofthefirstexperimentbyandTheoutcomesofthesecondexperimentbytheoutcomesforthetwoexperimentsaretheorderedpairsthesepairscanbeexhibitedastheentriesofanm×nrectangulararray,inwhichthepairisintheithrowandthejthcolumn.Therearemn

entriesinthisarray.Example2.18:Afamilyhasjustmovedtoanewcityandrequirestheservicesofbothanobstetricianandapediatrician.Therearetwoeasilyaccessiblemedicalclinics,eachhavingtwoobstetriciansandthreepediatricians.Thefamilywillobtainmaximumhealthinsurancebenefitsbyjoiningaclinicandselectingbothdoctorsfromthatclinic.Inhowmanywayscanthisbedone?DenotetheobstetriciansbyO1,O2,O3,andO4andthepediatriciansbyP1,…,P6.Thenwewishthenumberofpairs(Oi,Pj)forwhichOiandPjareassociatedwiththesameclinic.Becausetherearefourobstetricians,n1=4,andforeachtherearethreechoicesofpediatrician,son2=3.ApplyingtheproductrulegivesN=n1n2=12possiblechoices.TreediagramsO1O2O3O4P1P2P3P1P2P3P4P5P6P4P5P6AmoregeneralproductruleIftherearepexperimentsandthefirsthaspossibleoutcomes,thesecondandthepthpossibleoutcomes,thenthereareatotalofpossibleoutcomesforthepexperiments.

Example2.9:an8-bitbinarywordisasequenceof8digits,ofwhicheachmaybeeither0or1.Howmanydifferent8-bitwordsarethere?Therearetwochoicesforfirstbit,twoforthesecond,etc,andthusthereare2×2×2×2×2×2×2×2=28=256suchwords.PermutationsDefinition:Anyorderedsequenceofkobjectstakenfromasetofndistinctobjectsiscalledapermutation

ofsizekoftheobjects.Thenumberofpermutationsofsizekthatcanbeconstructedfromthenobjectsisdenotedby.Definition:Foranypositiveintegerm,m!isread“mfactorial”andisdefinedbysoCombinationsDefinition:Givenasetofndistinctobjects,anyunorderedsubsetofsizekoftheobjectsiscalledacombination.Thenumberofcombinationsofsizekthatcanbeformedfromthendistinctobjectswillbedenotedby.Example2.22Abridgehandconsistsofany13cardsselectedfroma52-carddeckwithoutregardtoorder.LetA={thehandconsistsentirelyofspadesandclubswithbothsuitsrepresented}B={thehandconsistsofexactlytwosuits}WhatistheprobabilityofeventAandeventB?Solution:

ExampleConsidertheexperimentofselectingfivecardsfromadeckof52cards.LetA={thefivecardsnumberdifferenteachother}B={twocardshavethesamenumber,andotherthreecardsalsohavehomologynumber}C={twocardshavethesamenumber,theotherthreecardshavedifferentnumber}D={fivecardshavefourkindsofdesigns}Example2.23Arentalcarservicefacilityhas10foreigncarsand15domesticcarswaitingtobeservicedonaparticularSaturdaymorning.BecausetherearesofewmechanicsworkingonSaturday,only6canbeserviced.Ifthe6arechosenatrandom,whatistheprobabilitythat3ofthecarsselectedaredomesticandtheother3areforeign?Whatistheprobabilitythatatleast3domesticcarsareselected?Solution:

Exercise1LetthesevenlettersC,C,E,E,I,N,Srankinaline,thentheprobabilityofjustarrangedawordSCIENCEis()(a)1/7!(b)2/7!(c)3/7!(d)4/7!Exercise2Twolettersarerandomlyputintofourpostboxes,thentheprobabilityoftheformertwopostboxeshavingnolettersis?dExercise3Therearetensameballswhichlabeled1,2,…,10inthebag.Nowthreeballsarerandomlyselectedfromthebag,andwriteddownthenumber.Calculatetheprobabilityof(1)theminimalnumberis5;(2)themaximumnumberis5.ExerciseP7438:

Aboxinacertainsupplyroomcontainsfour40-wlightbulbs,five60-wbulbs,andsix75-wbulbs.Supposethatthreebulbsarerandomlyselected.Whatistheprobabilitythatexactlytwooftheselectedbulbsarerated75-w?b.Whatistheprobabilitythatallthreeoftheselectedbulbshavethesamerating?c.Whatistheprobabilitythatonebulbofeachtypeisselected?d.Supposenowthatbulbsaretobeselectedonebyoneuntila75-wbulbisfound.Whatistheprobabilitythatitisnecessarytoexamineatleastsixbulbs?ExerciseP7539Fifteentelephoneshavejustbeenreceivedatanauthorizedservicecenter.Fiveofthesetelephonesarecellular,fivearecordless,andtheotherfivearecordedphones.Supposethatthesecomponentsarerandomlyallocatedthenumbers1,2,..,15toestablishtheorderinwhichtheywillbeserviced.a.Whatistheprobabilitythatallthecordlessphonesareamongthefirsttentobeserviced?b.Whatistheprobabilitythatafterservicingtenofthesephones,phonesofonlytwoofthethreetypesremaintobeserviced?c.Whatistheprobabilitythattwophonesofeachtypeareamongthefirstsixserviced?ExerciseP7542Threemarriedcoupleshavepurchasedtheaterticketsandareseatedinarowconsistingofjustsixseats.Iftheytaketheirseatsinacompletelyrandomfashion(randomorder),whatistheprobabilitythatJimandPaula(husbandandwife)sitinthetwoseatsonthefarleft?WhatistheprobabilitythatJimandPaulaendupsittingnexttooneanother?Whatistheprobabilitythatatleastoneofthewivesendsupsittingnexttoherhusband?Example:Therearemboysandngirls,theystandrandomlyonahorizontalline,LetA={girlsstandtogether}B={girlsstandtogether,andboysstandtogether,too}C={thereisatleastoneboybetweenanytwogirls}